Multi-digit+Subtraction

Multi-digit Subtraction: Many interesting alternative procedures
Here are alternatives that many students like, based on counting strategies they learn at an early age (compensating strategies):


 * 1.** For the problem 73 - 46, add 3 to 73 to get 76, then subtract 46, leaving 30. But then take away the 3 that you added earlier: 27.

73 - 46 76 - 46 = 30 30 **- 3** = 27
 * __+3__**


 * 2.** A "counting up" approach is to start with the lower number and add tens until you "overshoot," and then work backwards: 46 + 30 is 76, but it's 3 less: 30 - 3 = 27


 * 3.** An earlier "counting up" strategy also works: Add 4 to 46 to get 50, add 20 to 50 to get 70 (keep track, you've added 24 so far), than add three more to get 73: The answer is 27.


 * 4.** A "tens and ones" subtraction approach, which may be a stepping-stone to the traditional algorithm for children, starts by subtracting 40 (the tens) from 73, to get 33, then counting down 6 more: 32, 31, 30, 29, 28, 27.


 * 5**. A student might also subtract 50 from 73 to get 23, then recognizing that he or she subtracted too much, add 4 back to 23: 27.

Regrouping with unifix cubes and base ten blocks
To teach the traditional regrouping procedure, students need a firm grasp of place value, which results from using strategies like the ones above that involve skip counting up and down by tens, separately from counting by ones. They should model two-digit addition using unifix cubes first, where they physically stack 10 ones together to form a ten, and then base 10 blocks, where they trade 10 cubes for a rod.

They don't begin two-digit subtraction until after they have mastered at least one of the strategies above. Then they start with problems that don't require regrouping, e.g. Jackie has 45 crayons. She gives 23 crayons to Theo. How many crayons does she have left? Students count out 4 tens and 5 ones, then physically remove 2 tens and 3 ones, then count what's left over.

Then modify the original problem to make it a regrouping problem ("She gives 27 crayons to Theo"). Ask them to try this, using unifix cubes at first. Many children will recognize that they can break apart one of the tens to produce 10 ones, then continue with the subtraction. They might start by subtracting the tens first, then recognize that they don't have enough ones in the larger number to do the rest of the subtraction.

When students are comfortable with unifix cubes, they can change to base ten blocks.

Make sure that you record their work in symbols, to begin to translate to the written procedure.